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Degree sum of a pair of independent edges and Z3-connectivity
Ziwen HUANG,Xiangwen LI
Front. Math. China. 2016, 11 (6): 1533-1567.
https://doi.org/10.1007/s11464-015-0457-z
Let G be a 2-edge-connected simple graph on n vertices. For an edge e = uv ∈ E(G), define d(e) = d(u) + d(v). Let ℱ denote the set of all simple 2-edge-connected graphs on n≥4 vertices such that G ∈ ℱ if and only if d(e) + d(e')≥2n for every pair of independent edges e, e' of G. We prove in this paper that for each G ∈ ℱ, G is not Z3-connected if and only if G is one of K2,n−2, K3,n−3, K+2,n−2, K+3,n−3 or one of the 16 specified graphs, which generalizes the results of X. Zhang et al. [Discrete Math., 2010, 310: 3390–3397] and G. Fan and X. Zhou [Discrete Math., 2008, 308: 6233–6240].
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Numerical simulations for G-Brownian motion
Jie YANG,Weidong ZHAO
Front. Math. China. 2016, 11 (6): 1625-1643.
https://doi.org/10.1007/s11464-016-0504-9
This paper is concerned with numerical simulations for the GBrownian motion (defined by S. Peng in Stochastic Analysis and Applications, 2007, 541–567). By the definition of the G-normal distribution, we first show that the G-Brownian motions can be simulated by solving a certain kind of Hamilton-Jacobi-Bellman (HJB) equations. Then, some finite difference methods are designed for the corresponding HJB equations. Numerical simulation results of the G-normal distribution, the G-Brownian motion, and the corresponding quadratic variation process are provided, which characterize basic properties of the G-Brownian motion. We believe that the algorithms in this work serve as a fundamental tool for future studies, e.g., for solving stochastic differential equations (SDEs)/stochastic partial differential equations (SPDEs) driven by the G-Brownian motions.
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